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Chapter 9

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Chapter 9

Rotation of Rigid Bodies

- To describe rotation in terms of:
- angular coordinates (q)
- angular velocity (w)
- angular acceleration (a)

- To understand moment of inertia (I):
- how it depends upon rotation axes
- how it relates to rotational kinetic energy
- how it is calculated

- A wind turbine, a CD, a ceiling fan, and a Ferris wheel all involve rotating rigid objects.
- Real-world rotations can be very complicated because of stretching and twisting of the rotating body. But for now we’ll assume that the rotating body is perfectly rigid.

- Consider a meter with a needle rotating about a fixed axis.
- Angle (in radians) that needle makes with +x-axis is a coordinate for rotation.

- Example: A car’s speedometer needle rotates about a fixed axis.
- Angle the needle makes with negativex-axis is the coordinate for rotation.
- KEY: Define your directions!

55

45

35

25

15

q

- Angles in radians = s/r.
- One complete revolution is 360° = 2π radians.

- Angular displacement of a body is = 2 – 1 (in radians)
- Average angular velocityof a body is av = /t(in radians/second)

- The rotation axis matters!
- Subscript z means that the rotation is about the z-axis.
- Instantaneous angular velocity is z = d/dt.
- Counterclockwise rotation is positive;
- Clockwise rotation is negative.

- Flywheel diameter 0.36 m;
- Suppose q(t) = (2.0 rad/s3) t3

- Flywheel diameter 0.36 m; q = (2.0 rad/s3) t3
- Find q at t1 = 2.0 s and t2 = 5.0 s
- Find distance rim moves in that interval
- Find average angular velocity in rad/sec & rev/min
- Find instantaneous angular velocities at t1 & t2

- Angular velocity is defined as a vector whose direction is given by the right-hand rule.

- Average angular acceleration is avg-z = z/t.
- Instantaneous angular acceleration is z = dz/dt = d2/dt2.

- Average angular acceleration is avg-z = z/t.
- Instantaneous angular acceleration is z = dz/dt = d2/dt2.

For same flywheel with dia = 0.36 m; q(t) = (2.0 rad/s3) t3

find average angular accelerations between t1 & t2, & instantaneous accelerations at those t’s

- For a fixedrotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.

- For a fixedrotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.
- BUT THEY DON’T HAVE TO BE IN THE SAME DIRECTION!

w Speeds up!

w Slows down!

- Linear and Angular Motion with constant acceleration equations are very similar!

- A Blu-ray disc is coming to rest after being played.
- @ t = 0, w = 27.5 rad/sec; a = -10.0 rad/s2
- What is w at t = 0.3 seconds?
- What angle does PQ make with x axis then?

- For a point a distance r from the axis of rotation:
its linear speed is v = r (meters/sec)

- For a point a distance r from the axis of rotation:
its linear tangential acceleration is atan = r (m/s2)

its centripetal (radial) acceleration is arad = v2/r = r

- Whirl discus in circle of r = 80 cm; at some time t athlete is rotating at 10.0 rad/sec; speed increasing at 50.0 rad/sec/sec.
- Find tangential and centripetal accelerations and overall magnitude of acceleration

- Say rotation of propeller is at a constant 2400 rpm, as plane flies forward at 75.0 m/s at constant speed.
- But…tips of propellers must move slower than 270 m/s to avoid excessive noise.
- What is maximum propeller radius?
- What is acceleration of the tip?

- Tips of propellers must move slower than than 270 m/s to avoid excessive noise. What is maximum propeller radius? What is acceleration of the tip?

How much force it takes to get something rotating, and how much energy it has when rotating, depends on WHERE the mass is in relation to the rotation axis.

- Getting MORE mass, FARTHER from the axis, to rotate will take more force!
- Some rotating at the same rate with more mass farther away will have more KE!

- The moment of inertia of a set of discrete particles is
- I = m1r12 + m2r22 + … = miri2
- Rotational kinetic energy of rigid body with moment of inertia IKE(rotation) = 1/2 I2 (still in Joules!)
- Since I varies by location, KE varies depending upon axis!

- What is I about A?
- What is I about B/C?
- What is KE if it rotates through A with w=4.0 rad/sec?

- Wrap a light, non-stretching cable around solid cylinder of mass 50 kg; diameter .120 m. Pull for 2.0 m with constant force of 9.0 N. What is final angular speed and final linear speed of cable?

Consider falling mass “m” tied to rotating wheel of mass Mand radius R

What is the resulting speed of the small mass when it reaches the bottom?

Consider falling mass “m” tied to rotating wheel of mass Mand radius R

What is the resulting speed of the small mass when it reaches the bottom?

Method 1: Energy!

mgh

½ mv2+ ½ Iw2

mgh = ½ mv2+ ½ Iw2

I = ½ MR2 for disk

mgh = ½ (m+ ½ M) v2

V = [2gh/(1+M/2m)] ½

- What happens if you rotate about an EXTERNAL axis, not internal?
- Spinning planet orbiting around the Sun
- Rotating ball bearing orbiting in the bearing
- Mass on turntable

- What happens if you rotate about an EXTERNAL axis, not internal?
- Effect is a COMBINATION of TWO rotations
- Object itself spinning;
- Point mass orbiting

- Net rotational inertia combines both:
- The parallel-axis theorem is: IP = Icm + Md2

- The parallel-axis theorem is: IP = Icm + Md2.

M

d

- The parallel-axis theorem is: IP = Icm+Md2.
- Mass 3.6 kg, I = 0.132 kg-m2 through P. What is I about parallel axis through center of mass?